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The final state problem for the nonlinear Schrodinger equation in dimensions 1, 2 and 3

Andrew Hassell, Qiuye Jia

Abstract

In this article we consider the defocusing nonlinear Schrödinger equation, with time-dependent potential, in space dimensions $n=1, 2$ and $3$, with nonlinearity $|u|^{p-1} u$, $p$ an odd integer, satisfying $p \geq 5$ in dimension $1$, $p \geq 3$ in dimension $2$ and $p=3$ in dimension $3$. We also allow a metric perturbation, assumed to be compactly supported in spacetime, and nontrapping. We work with module regularity spaces, which are defined by regularity of order $k \geq 2$ under the action of certain vector fields generating symmetries of the free Schrödinger equation. We solve the large data final state problem, with final state in a module regularity space, and show convergence of the solution to the final state.

The final state problem for the nonlinear Schrodinger equation in dimensions 1, 2 and 3

Abstract

In this article we consider the defocusing nonlinear Schrödinger equation, with time-dependent potential, in space dimensions and , with nonlinearity , an odd integer, satisfying in dimension , in dimension and in dimension . We also allow a metric perturbation, assumed to be compactly supported in spacetime, and nontrapping. We work with module regularity spaces, which are defined by regularity of order under the action of certain vector fields generating symmetries of the free Schrödinger equation. We solve the large data final state problem, with final state in a module regularity space, and show convergence of the solution to the final state.

Paper Structure

This paper contains 25 sections, 27 theorems, 222 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Nonlinear Schrödinger equation and the final state problem
  3. Main result
  4. Related literature
  5. Links with the articles gell2022propagation and gell2023scattering
  6. Outline of the proof, and structure of the paper
  7. 1-cusp structure
  8. Module regularity spaces
  9. Module regularity spaces and their properties
  10. A Gagliardo-Nirenberg type inequality for Module regularity spaces
  11. $L^p$ estimates
  12. The contraction mapping scheme
  13. The integral equation formulation and Strichartz-type estimate
  14. The contraction mapping argument
  15. Convergence
  16. ...and 10 more sections

Key Result

Theorem 1.1

Suppose that $n = 1,2$ or $3$, that $p$ satisfies eq:np, and that $k \geq 2$. Let the potential function $V(t, z)$ be in $\langle t \rangle^{-1} L^\infty_t \mathcal{W}^k_{\mathcal{M}_{t,0}}$, and let $f_+ \in \mathcal{W}^k_{\mathcal{N}}$ be final state data at $t = +\infty$. Then there is a unique s In particular this implies pointwise convergence, since $\mathcal{W}^k_{\mathcal{N}}$ embeds into $

Figures (1)

  • Figure 1: The radially compactified spacetime, with time axis oriented vertically. The boundary, which is an $S^n$, is referred to as spacetime infinity, and consists of an open 'northern hemisphere', where $t = +\infty$, an open 'southern hemisphere', where $t = -\infty$, and the equator, which is the indicated great circle. Any time slice $t = \mathrm{constant}$ meets spacetime infinity at the equator. The smooth function $(1 + t^2 + |z|^2)^{-1/2}$ defines (i.e. vanishes simply at) spacetime infinity. The function $\zeta := z/(2t)$ extends to a smooth coordinate either on the open northern hemisphere or the open southern hemisphere. It is infinite at the equator.

Theorems & Definitions (60)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Definition 3.1
  • Proposition 3.2
  • ...and 50 more